The proof of Theorem 3.10 in Introduction to Analytic Number Theory by Apostol goes like this:

Theorem 3.10If $ h = f * g $, let

$$

H(x) = sum_{n le x} h(n),

F(x) = sum_{n le x} f(n), text{ and }

G(x) = sum_{n le x} g(n). $$

Then we have

$$

H(x)

= sum_{n le x} f(n) G left( frac{x}{n} right)

= sum_{n le x} g(n) F left( frac{x}{n} right). $$PROOF. We make use of the associative law (Theorem 2.21) which relates the operations $ circ $ and $ * $. Let

$$

U(x) = begin{cases} 0 & text{ if } 0 < x < 1, \

1 & text{ if } x ge 1. end{cases}

$$

Then $ F = f circ U $, $ G = g circ U $, and we have

$$ f circ G = f circ (g circ U) = (f * g) circ U = H, $$

$$ g circ F = g circ (f circ U) = (g * f) circ U = H. $$

This completes the proof.

I am curious to understand why use generalized convolution and Dirichlet multiplication here at all.

Since section 3.5, the book has been relying on the following reordering of summation at various places,

$$

sum_{n le x} sum_{d | n} f(d)

= sum_{substack{q, d \ qd le x}} f(d)

= sum_{d le x} sum_{q le frac{x}{d}} f(q).

$$

Can we not arrive at a simpler proof with a similar reordering of summation like this:

begin{align*}

H(x)

& = sum_{n le x} h(n)

= sum_{n le x} (f * g)(n)

= sum_{n le x} sum_{d | n} f(d) gleft(frac{n}{d}right)

= sum_{substack{q, d \ qd le x}} f(d) gleft(qright) \

& = sum_{d le x} sum_{q le frac{x}{d}} f(d) g(q)

= sum_{d le x} f(d) sum_{q le frac{x}{d}} g(q)

= sum_{n le x} f(n) sum_{q le frac{x}{n}} g(q)

= sum_{n le x} f(n) Gleft(frac{x}{n}right).

end{align*}

I am trying to understand if a simpler proof using basic arithmetic and reordering of summations is correct or if I am missing something and there is a specific reason why Apostol went for a proof using generalized convolution and Dirichlet multiplication.